This is Prim's algorithm [25,8,27,15] for solving the minimum
spanning tree problem for an undirected graph with weighted edges. A
MST is a set of edges that connects all the vertices in the graph
where the total weight of the edges in the tree is minimized. See
Section Minimum
Spanning Tree Problem for more details. The implementation is
simply a call to dijkstra_shortest_paths()
with the appropriate choice of comparison and combine functors.
The pseudo-code for Prim's algorithm is listed below.
The algorithm as implemented in Boost.Graph does not produce correct results on
graphs with parallel edges.

Where Defined

Parameters

An undirected graph. The type Graph must be a
model of Vertex List Graph
and Incidence Graph. It should not
contain parallel edges.Python: The parameter is named graph.

OUT: PredecessorMap p_map

The predecessor map records the edges in the minimum spanning
tree. Upon completion of the algorithm, the edges
(p[u],u) for all u in V are in the minimum spanning
tree. If p[u] = u then u is either the root of the
tree or is a vertex that is not reachable from the root.
The PredecessorMap type must be a Read/Write
Property Map
with key and vertex types the same as the vertex descriptor type
of the graph.Python: Must be a vertex_vertex_map for the graph.

Named Parameters

IN: root_vertex(vertex_descriptor r)

The vertex that will be the root of the minimum spanning tree.
The choice of the root vertex is arbitrary.Default:*vertices(g).first

IN: weight_map(WeightMap w_map)

The weight or ``length'' of each edge in the graph.
The type WeightMap must be a model of
Readable Property Map. The edge descriptor type of
the graph needs to be usable as the key type for the weight
map. The value type for the map must be
the same as the value type of the distance map, and that type must be Less Than
Comparable.Default:get(edge_weight, g)Python: Must be an edge_double_map for the graph.Python default: graph.get_edge_double_map("weight")

IN: vertex_index_map(VertexIndexMap i_map)

This maps each vertex to an integer in the range [0,
num_vertices(g)). This is necessary for efficient updates of the
heap data structure when an edge is relaxed. The type
VertexIndexMap must be a model of
Readable Property Map. The value type of the map must be an
integer type. The vertex descriptor type of the graph needs to be
usable as the key type of the map.Default:get(vertex_index, g)
Note: if you use this default, make sure your graph has
an internal vertex_index property. For example,
adjacency_list with VertexList=listS does
not have an internal vertex_index property.
Python: Unsupported parameter.

UTIL/OUT: distance_map(DistanceMap d_map)

The weight of the spanning tree edge into each
vertex in the graph g is recorded in this property map, with edges
directed away from the spanning tree root.
The type DistanceMap must be a model of Read/Write
Property Map. The vertex descriptor type of the
graph needs to be usable as the key type of the distance map, and the value
type needs to be the same as the value type of the weight_map
argument.Default:iterator_property_map created from a
std::vector of the WeightMap's value type of size
num_vertices(g) and using the i_map for the index
map.Python: Must be a vertex_double_map for the graph.

UTIL/OUT: color_map(ColorMap c_map)

This is used during the execution of the algorithm to mark the
vertices. The vertices start out white and become gray when they are
inserted in the queue. They then turn black when they are removed
from the queue. At the end of the algorithm, vertices reachable from
the source vertex will have been colored black. All other vertices
will still be white. The type ColorMap must be a model of
Read/Write
Property Map. A vertex descriptor must be usable as the key type
of the map, and the value type of the map must be a model of
Color Value.Default: an iterator_property_map created from a std::vector
of default_color_type of size num_vertices(g) and
using the i_map for the index map.Python: The color map must be a vertex_color_map for
the graph.

OUT: visitor(DijkstraVisitor v)

Use this to specify actions that you would like to happen
during certain event points within the algorithm.
The type DijkstraVisitor must be a model of the
Dijkstra Visitor concept.
The visitor object is passed by value [1].Default:dijkstra_visitor<null_visitor>Python: The parameter should be an object that derives from
the DijkstraVisitor type
of the graph.

Complexity

The time complexity is O(E log V).

Example

Notes

[1]
Since the visitor parameter is passed by value, if your visitor
contains state then any changes to the state during the algorithm
will be made to a copy of the visitor object, not the visitor object
passed in. Therefore you may want the visitor to hold this state by
pointer or reference.